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\(C\)-totally real submanifolds of \(\mathbb{R}^{2n+1}\) satisfying a certain inequality. (English) Zbl 1029.53064

In theorem 1 the author establishes a sharp inequality between the squared mean curvature \(||H||^2\) and the scalar curvature \(\tau\) for a \(C-\)totally real submanifold of maximum dimension in a Sasakian space form. He also investigates the equality case in the following:
Theorem 2. Let \(i:M^n\rightarrow \mathbb{R}^{2n+1}\) be a \(C\)-totally real immersion satisfying the equality \(||H||^2={{2(n+2)}\over{n^2(n-1)}}\tau\). Then either \(M\) is a totally geodesic submanifold and hence locally isometric to the real space \(\mathbb{R}^n \) or the set \(U\) of non-totally geodesic points in \(m\) is a dense subset of \(M\), \(U\) is an open portion of a Whitney sphere \(\widetilde w(S^n)\) with \(a>1\) and, up to rigid body motions of \(\mathbb{R}^{2n+1}\), the immersion \(i\) is given by \(\widetilde w\), where \(\widetilde w:S^n\rightarrow \mathbb{R}^{2n+1}\) is the immersion lifted from the Whitney immersion. The map \(f:E^{n+1}\rightarrow \mathbb{C}^{n}\) from Euclidean space into the complex Euclidean space \(\;f(x_0,x_1,\dots,x_n)={{1}\over{1+x_0^2}}(x_1,\dots,x_n,x_0x_1,\dots,x_0x_n)\) restricted on the unit hypersphere \(S^n\) is called the Whitney immersion.

MSC:

53C40 Global submanifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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